The proof of Theorem 16 in [1]} directly establishes the fact that under the conditions of the lemma, we may replace all occurrences of some \(w_i\)  consistently with any other word \(u\)  and still have a solution to the word equation. By ensuring that we swap \(w_i\)  for a word \(u\)  of the same length, we ensure that the new solution satisfies the length constraints whenever the old one does. It follows that the new word also belongs to \(L\)  so the statement of the lemma holds.
